A Pause for Reflection — Question 6

Question 6

Combining independent mass measurements gives a precision-weighted estimate

Two independent Gaussian measurements give $m_1=100\pm2$ and $m_2=106\pm3$ MeV/$c^2$. Neglect common systematic errors.

1. Compute the inverse-variance weighted mean and uncertainty.

2. Find the discrepancy between the two measurements in units of its standard deviation, and explain why this is a consistency diagnostic.

Question 6 - Solution

Combining independent mass measurements gives a precision-weighted estimate

(a) Method and calculation

$$\bar m=\frac{100/4+106/9}{1/4+1/9}=\boxed{101.846\ {\mathrm{MeV}}/c^2}.$$$$\boxed{\sigma_{\bar m}=(1/4+1/9)^{-1/2}=1.66410\ {\mathrm{MeV}}/c^2}.$$

(b) Interpretation and further calculation

$$\boxed{z=(106-100)/\sqrt{2^2+3^2}=1.66410}.$$$$\chi^2_{\min}=z^2=\boxed{36/13=2.76923}\quad(1\text{ degree of freedom}).$$

Physical check

The combination assumes a shared true mass and independent errors; a common calibration uncertainty would not average down this way.